Questions tagged [hilbert-schemes]
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196 questions
21
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$8$-ary operation $(\mathbb{P}^2)^8 \text{ }-\to \mathbb{P}^2$, can we say anything about what this formula would look like?
My friend, who is currently taking an algebraic geometry course from an unnamed prolific poster on MO, told me about the following bonus question on one of his problem sets a few weeks ago.
...
user61522
20
votes
3
answers
2k
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Is there a scheme parametrizing the closed subgroups of an algebraic group?
In the following, let $G=\operatorname{GL}_n(\mathbb{C})$ or $G=\operatorname{\mathbb PGL}_n(\mathbb{C})$, depending on whichever has a better chance of yielding an affirmative answer. One could more ...
- 4,902
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2
answers
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Why do flag manifolds, in the P(V_rho) embedding, look like products of P^1s?
Bert Kostant mentioned an odd fact to me some time ago. As usual (with such statements), fix a
complex, connected, reductive) Lie group $G$, with maximal torus $T$, and Weyl vector $\rho$ equal to ...
- 27.8k
20
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Fuss-Catalan algebras and non-commutative Hilbert schemes
Hello, this is a question regarding Reineke's paper "Cohomology of non-commutative Hilbert schemes", http://arxiv.org/abs/math/0306185, and more precisely the formula on page 4 there (at $n=1$), ...
- 1,363
16
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1
answer
1k
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Reference Request for Hilbert Schemes
I'm a physicist working on Fractional Quantum Hall effect. The mathematical subjects of study are symmetric, translational invariant, homogeneous polynomials on $\mathbb{C}$. Very early in my study I ...
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5
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When are Hilbert schemes smooth?
I know that Hilbert schemes can be very singular. But are there any interesting and nontrivial Hilbert schemes that are smooth? Are there any necessary conditions or sufficient conditions for a ...
- 21k
14
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1
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822
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Counting Hilbert polynomials of projective varieties
EDIT. Fix $n,d,k\in\mathbb{N}$. Let us consider the set $\mathcal{P}_{n,d,k}$ of polynomials $P$ in one variable for which there exits a closed irreducible subvariety $X_P\subset \mathbb{C}\mathbb{P}^...
- 21.8k
13
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3
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1k
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Reference for combinatorics of cell decomposition of the Hilbert scheme of points in the plane
It is known from either Morse theory or Bialynicki-Birula decomposition that the fixed points of a ${\mathbb{C}}^*$ action on a smooth algebraic variety over $\mathbb{C}$ determine a cell ...
- 2,964
13
votes
1
answer
553
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On non-representability of certain hom schemes
Let $k$ be an algebraically closed field. It is well-known that the isom-sheaf Isom$(\mathbb{A}^1_k,\mathbb{A}^1_k)$ is not representable by an algebraic space. (To be clear, the functor Isom$(\mathbb{...
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2
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893
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Are spaces of holomorphic maps manifolds?
Hello,
Let $X$ and $Y$ be two smooth (probably projective) algebraic varieties defined over $\mathbf{C}$.
What is known in general about the (topological) space of holomorphic maps $\mathrm{Hol}(X(\...
- 2,521
12
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1
answer
2k
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Hilbert schemes and moduli of ideal sheaves
Let $X$ be a smooth projective variety over $\mathbb{C}$. The Hilbert scheme on $X$ parametrizes quotients $\mathcal{O}_X \to E$ with fixed Hilbert polynomial. Let us fix the Hilbert polynomial to ...
- 783
11
votes
2
answers
1k
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Parameter space for complete intersections and their discriminant
Consider globally complete intersections in $\mathbb{P}^n$, of codimension $k$, of some fixed multi-degree $(d_1,\dots,d_k)$.
Is there some nice (i.e. "explicit") parameter space for them?
(even if ...
- 2,232
11
votes
1
answer
2k
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Algebraic cycles, Chow spaces and Hilbert-Chow morphisms
In the sequel, let $S$ be a scheme, and $X$ a locally of finite type algebraic space over $S$.
In his thesis ([R1-R4]), David Rydh introduces, among several others, the notion of relative cycles on $...
user87684
11
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1
answer
737
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Chow ring of Hilbert scheme of 4 points in $\mathbb{P}^2$
What is known about the Chow ring of the Hilbert scheme of length 4 subschemes of $\mathbb{P}^2$?
I know there is work on cycles on Hilbert schemes in the literature, but I don't know what can be ...
- 1,537
10
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3
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2k
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Families of ideal sheaves: What's the correct definition?
I'm looking at Bridgeland's paper "Flops and Derived categories" and I got confused on what he meant by a family of ideal sheaves.
Let $Y$ be a scheme, and let $S$ be another scheme. A ...
- 3,806
10
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4
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1k
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Moduli spaces in applied mathematics and condensed matter physics?
In this MO question it is stated that there is a relation between some aspects of condensed matter physics (namely the fractional quantum Hall effect) and the algebraic geometry of Hilbert schemes.
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10
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2
answers
625
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Why is Maps(X,Y) an open subfunctor of Hilb(X x Y)?
Let $X$ and $Y$ be projective schemes. Then we can define the mapping scheme between them, $\rm{Maps}(X,Y)$ as follows:
To any map $f:X\rightarrow Y$ we consider the graph $\Gamma_f$ as a closed ...
- 5,548
10
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0
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405
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Hilbert schemes of points on toric surfaces
Let $\mathrm{S}$ be a smooth toric surface. The Hilbert scheme of $n$ points $\mathrm{Hilb}^n(\mathrm{S})$ inherits a torus action, but need not admit the structure of a toric variety itself. For ...
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3
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2k
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Reference request: is the punctual Hilbert scheme irreducible?
The punctual Hilbert scheme in dimension $d$ parameterizes ideals $I$ of codimension $n$ in $k[x_1,\dots, x_d]$ which are contained in some power of the ideal $(x_1,\dots, x_d)$. In other words, it is ...
9
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1
answer
713
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There are only finitely many varieties up to deformation
Let $h$ be a polynomial. Then results of several authors (including Chow, Grothendieck, Matsusaka, Mumford, Kollar and Viehweg) imply that the moduli space of polarized varieties with Hilbert ...
- 93
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0
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387
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Kähler metric on the Hilbert scheme of points on a surface
Question. Let $S$ be a non-singular complex projective surface and let $S^{[n]}$ be its Hilbert scheme of $n$ points. Is there a natural way to associate to a Kähler metric $\omega$ on $S$ a Kähler ...
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2
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1k
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Relationship between Hilbert schemes and deformation spaces
Hi, I'm just starting to learn about deformation theory (via Hartshorne's Deformation theory, as well as Fantechi's section of FGA explained), and I feel like I'm confused about fundamental concepts. ...
- 371
8
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1
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285
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Degrees of syzygies of points in $\mathbb P^2$
Let $X$ be a collection of points in $\mathbb P^2$ over the complex numbers. Let $I_X$ be the defining ideal. I am interested in knowing when:
The syzygies of $I_X$ contains no linear forms. Since ...
- 30.5k
8
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0
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469
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Hilbert scheme of projectively normal elliptic curves
Consider the Hilbert scheme of degree $n$, genus $1$ curves in $\mathbb P^{n-1}$. It contains the locus of smooth curves embedded by the complete linear system of a degree $n$ divisor. Let $X_n$ be ...
- 148k
7
votes
3
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652
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Irreducible "family" of relative effective divisors of a smooth morphism
Let $\pi:X\rightarrow Y$ be a smooth proper (assume projective if needed) morphism of schemes with $Y$ locally noetherian, and let $Z\subset X$ be an irreducible integral closed subscheme containing ...
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7
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1
answer
506
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Is there a way to check if a relative Hilbert Scheme is reduced?
More specifically, suppose I have a rational curve on a complete intersection, and I know that the relative Hilbert Scheme is not smooth at the point corresponding to my rational curve. Is there any ...
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7
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0
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178
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Is the universal object over a Hilbert scheme connected?
Hartshorne proved in his thesis that if $S$ is connected, then the Hilbert scheme $\operatorname{Hilb}^p=\operatorname{Hilb}^p(\mathbb{P}^n_S/S)$ is too (where $p\in \mathbb{Q}[z]$). Can the same be ...
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7
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0
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566
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Philosophical underpinnings of Grothendieck's construction of the Hilbert scheme
Long ago when I was in grad school I was told that Grothendieck's construction of the Hilbert scheme is rooted in two main technical points: Castelnuovo-Mumford regularity and Mumford flattening ...
- 2,964
7
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0
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188
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Open subfunctor of Quot Functor induced by open immersion
Let $f: X \rightarrow S$ be a morphism of noetherian schemes and $\mathfrak{Q}uot_{\mathcal{E}/X/S}$ be the functor parametrizing families of quotients of $\mathcal{E}$ in the category of locally ...
- 171
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0
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366
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Equivariant Hilbert schemes of points
Let $G$ be a finite subgroup of $\mathrm{SL}(2, \mathbb{C})$ and let $X$ be the quotient surface $X = \mathrm{Spec}(\mathbb{C}[x, y]^{G})$. Denote by $\mathrm{Hilb}^r([X])$ the equivariant Hilbert ...
- 143
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0
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293
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Status of Haiman's conjectures on the Isospectral Hilbert Scheme for dim X>2?
Let $X$ be a variety of arbitrary dimension, let $H$ denote the main component of the Hilbert scheme of points of $X$ (i.e, the closure of locus of reduced subschemes), and let $Z$ be the reduced ...
- 5,958
7
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0
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421
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Punctual Hilbert Schemes
Let $H$ be the Hilbert scheme of Artin local rings (quotients of a power series ring $R$ in $e$ variables over $\mathbb{C}$) of length $n$. Consider the set $G\subset H$ of rings $A$ with the property ...
- 6,262
6
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1
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338
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Deformation of curves and closed immersions
Let $\pi:\mathcal{C} \to B$ be a (flat) family of complex projective schemes of pure dimension $1$ with fixed Hilbert polynomial, in particular, for some $n \ge 3$, $\mathcal{C} \hookrightarrow \...
- 2,126
6
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0
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405
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This sum over partitions has unexpectedly nice denominators
Fix an integer $n\geq0$, a power series $\gamma \in \mathbb Q[[X]]$ with valuation 1, and a symmetric function $f$ (with coefficients in $\mathbb Q$). Now, consider the series
$$
S_n = \sum_{\Lambda\...
- 1,509
6
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0
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152
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A conjecture about sums over partitions arising from Hilbert scheme of points
$\DeclareMathOperator{\leg}{\operatorname{leg}}\DeclareMathOperator{\arm}{\operatorname{arm}}$The following situation arose from the study of some localization computations on Hilbert schemes of ...
- 1,509
6
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0
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168
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How does the "todd class operator" commute with Nakajima's q operators on Hilbert schemes of points on surfaces
Let $S$ be a surface, and $S^{[n]}$ the Hilbert scheme of $n$ points on $S$. One defines
$$
\mathbb H = \bigoplus_n H^*(S^{[n]}).
$$
One has Nakajima's operators $\mathfrak q_n(\alpha)$ (with $\alpha \...
- 1,509
6
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0
answers
170
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A relation of convergence in Hilbert scheme to convergence in sense of currents
Let $\{X_i\}$ be a sequence of closed irreducible $k$-dimensional subvarieties of $\mathbb{C}\mathbb{P}^n$ of degree $d$ (they may be assumed to be smooth if necessary). Assume that this sequence ...
- 21.8k
6
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0
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320
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A question on infinitesimal deformation (related to intersection theory)
Let $X$ be a connected projective scheme in $\mathbb{P}^n$. Assume, $2 \le \dim X \le n-2$. Let $H$ be a general hyperplane in $\mathbb{P}^n$. Denote by $Z:=X.H$ and $Z'=X.H^m$ for $m \gg 0$. Then ...
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2
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268
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Branch locus of a 6:1 cover of the grassmannian G(1,3)
Given a general quartic surface $S$ in $\mathbf{P}^3$, there is a natural 6:1 surjective map
$\phi: Hilb^2(S) \to G(1,3)$ sending $\{P,Q\}$ to the line through them in $\mathbf{P}^3$.
Can you ...
- 453
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1
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357
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If $X$ is a degree 3 smooth integral surface in $P^N$, $N > 3$, is it still true that it contains 27 lines?
It is a well known fact that a smooth cubic surface in $P^3$ contains 27 lines. One proof proceeds by moving through the parameter space $U$ of smooth cubics until one reaches an cubic that can be ...
- 1,462
5
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1
answer
1k
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Is projective morphism with projective fiber flat?
Let $X, Y$ be quasi-projective Noetherian schemes and $f:X \to Y$ be a projective surjective morphism. Assume that every fiber of $f$ is isomorphic to a projective space $\mathbb{P}^n$ for a fixed $n$....
- 1,593
5
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1
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Examples of nice reduced singularities on Hilbert schemes--Edited
In his "Murphy's Law" paper, Vakil showed that every "singularity type" (with a precise meaning) occurs on certain Hilbert schemes; for instance, the Hilbert scheme of nonsingular curves in projective ...
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2
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461
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Connectedness of Quot schemes
Let $X$ be a connected projective scheme over $\mathbb{C}$ and $E$ a coherent sheaf on $X$. Consider the Quot scheme $\operatorname{Quot}_X(E,P)$ of quotients of $E$ of fixed Hilbert polynomial $P$. ...
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1
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516
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Smooth quadric hypersurface, Hilbert scheme is blowup of Grassmannian?
Let $Q \subset \mathbb{P}^n$ be a smooth quadric hypersurface. Where can I find a proof of/can anyone supply a proof of$$\text{Hilb}_{2m + 1}(Q) \cong \text{Bl}_{OG(3, n+1)}G(3, n+1)?$$Can we conclude ...
- 51
5
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1
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831
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Hilbert scheme of points on a surface
Let $X$ be a complex surface and $X^{[n]}$ be the Hilbert scheme of finite analytic subspaces $Z$ for which $dimH^0(Z,\mathcal{O}_Z)=n$. I have trouble understanding $X^{[n]}$. That's what i've worked ...
- 53
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1
answer
193
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Multiple of a flat family of subschemes is flat
Let $X$ be a fixed curve (e.g. a Noetherian, projective scheme of dimension 1, of finite type over an algebraically closed field $k$) and let $S$ be an arbitrary parameter scheme over $k$. Let $D \...
- 479
5
votes
1
answer
164
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Connectedness of sub-varieties via Hilbert polynomials
Let $X$ be a sub-variety of $\mathbb CP^n$ and let $p_X(k)$ be its Hilbert polynomial. It is well known that some basic invariants of $X$ (such as its dimension) can be read from $p_X(k)$. I am ...
- 14.3k
5
votes
1
answer
350
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What are the "special" strata of Sym^n(C^2)?
The affine variety $Sym^n(\mathbb{C}^2)$ has a natural quantization as a spherical rational Cherednik algebra. Thus, any primitive ideal of the rational Cherednik algebra has an corresponding ideal ...
- 44.7k
5
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1
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504
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Construction of an atlas for the moduli stack $\mathcal{Bun}_X^{n,d}$ in F. Neumann's 'Algebraic Stacks and Moduli of Vector Bundles'
I'm reading Frank Neumann's "Algebraic Stacks and Moduli of
Vector Bundles" and have some problems to understand
a construction from the proof of:
Theorem 2.67. (page 81) The moduli stack $...
- 6,018
5
votes
1
answer
437
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A question on the morphism between Hilbert schemes
Let $L_1,L_2$ be two irreducible component of two different Hilbert schemes parametrizing closed subscheme in $\mathbb{P}^n$ and $\mathbb{P}^{n-1}$, respectively. Denote by $\pi_1: \mathcal{X}_1 \to ...
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